Svartholm [1]}
and Erdélyi [2]}, [3]} have supposed that the coefficients of their expansions in series of hypergeometric functions
satisfy recurrence relations having
the form given in (REF ).
Such conjecture has been repeated in subsequent literature [4]}, [5]}, [6]}, , including Erdélyi himself [7]}. However, there are three forms for the recurrence relations, namely,
\(&\begin{array}{l}\alpha _{0}b_{1}+\beta _{0}b_{0}=0,\qquad \alpha _{n}b_{n+1}+\beta _{n}b_{n}+\gamma _{n}b_{n-1}=0,\quad (n\ge 1),\end{array}\vspace{8.53581pt}\\&\begin{array}{r}{\left\lbrace \begin{array}{ll}\alpha _{0}b_{1}+\beta _{0}b_{0}=0,\quad \alpha _{1}b_{2}+\beta _{1}b_{1}+\left[\gamma _{1}+\alpha _{-1}\right]b_{0}=0,\quad \vspace{5.69054pt}\\\alpha _{n}b_{n+1}+\beta _{n}b_{n}+\gamma _{n}b_{n-1}=0,\quad (n\ge 2),\end{array}\right.}\end{array}\vspace{8.53581pt}\\&\begin{array}{l}\alpha _{0}b_{1}+\left[\beta _{0}+\alpha _{-1}\right]b_{0}=0,\qquad \alpha _{n}b_{n+1}+\beta _{n}b_{n}+\gamma _{n}b_{n-1}=0,\qquad (n\ge 1)\end{array}\) 
